Geometric Optimal Control with Applications

Accelerated Graduate CourseInstitute of Mathematics for Industry, Kyushu University,

Bernard BonnardInria Sophia Antipolis et Institut de Mathematiques de Bourgogne

9 avenue Savary

e21078 Dijon, France

Monique Chyba2565 McCarthy the Mall

Department of Mathematics

University of Hawaii

Honolulu, HI 96822, USA

with the help of

Gautier Picot, Aaron Tamura-Sato, Steven Brelsford

June-July 2015

Contents

2 Controllability 22.1 Notation from Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Controllability with Piecewise Constant Controls . . . . . . . . . . . . . . . . . . . . . 32.1.2 Integrating Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Nagano-Sussman Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.5 C∞-Counter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.6 Nonlinear Controllability and Chow Theorem . . . . . . . . . . . . . . . . . . . . . . . 62.1.7 Poisson Stability and Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.8 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.9 Controllability and Enlargement Technique . . . . . . . . . . . . . . . . . . . . . . . . 82.1.10 Evaluation of the Accessibility Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.11 Legendre-Clebsch Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.12 The Multi-Inputs Case: Goh Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.13 The Concept of Rigidity - Strong Legendre Clebsch and Goh Conditions as Necessary

Conditions for Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Acknowledgments 16

1

Chapter 2

Controllability

Higher order necessary optimality conditions for singular extremals are due to Goh, Kelley, and others; seethe survey article [10]. The higher order maximum principle is due to Krener [21] and Hermes [15]. The en-largement technique exists in a heuristic form in Hirschorn [16], but was conceptualized by Jurdjevic-Kupka[18] and fully exploited to get controllability conditions for right-invariant systems on Lie groups. The prob-lem of rigidity is everywhere-present in the literature about the abnormal problem in calculus of variations,see for instance Bliss [3]. It also appears in the article [4] in control theory and the neat analysis concerningthis problem is due to [1].

Lie brackets play a crucial role in analyzing the controllability properties of nonlinear control systems, andthe regularity properties of optimal trajectories.

2.1 Notation from Differential Geometry

We denote by M a smooth (C∞ or Cω) manifold of dimension n, connected and second countable. Wedenote by TM the fiber bundle and by T ∗M the cotangent bundle. Let V (M) be the set of smooth vectorfields on M and Diff(M) the set of smooth diffeomorphisms.

Definition 1. Let X ∈ V (M) and let f be a smooth function on M . The Lie derivative is defined as:LXf = df(X). If X,Y ∈ V (M), the Lie bracket is given by

ad X(Y ) = [X,Y ] = LY ◦ LX − LX ◦ LY .

If x = (x1, · · · , xn) are a local system of coordinates we have:

X(x) =n∑

i=1

Xi(x)∂

∂xi

LXf(x) =∂f

∂xX(x)

[X,Y ](x) =∂X

∂x(x)Y (x)− ∂Y

∂x(x)X(x)

The mapping (X,Y ) 7→ [X,Y ] is R-linear and skew-symmetric. Moreover, the Jacobi identity holds:

[X, [Y, Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0.

Definition 2. Let X ∈ V (M). We denote by x(t, x0) the maximal solution of the Cauchy problem x(t) =X(x(t)), x(0) = x0. This solution is defined on a maximal open interval J containing 0. We denote byexp tX the local one parameter group associated to X, that is: exp tX(x0) = x(t, x0). The vector field X issaid to be complete if the trajectories can be extended over R.

Definition 3. Let X ∈ V (M) and φ ∈ Diff(M). The image of X by φ is φ ∗X = dφ(X ◦ φ−1).

2

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We recall the following results.

Proposition 1. if X,Y ∈ V (M) and φ ∈ Diff(M), we have:

1. The one parameter local group of Z = −φ ∗X is given by:

exp tZ = φ ◦ exp tX ◦ φ−1

2. φ ∗ [X,Y ] = [φ ∗X,φ ∗ Y ]

3. The Baker-Campbell-Hausdorf (BCH) formula is:

exp sX exp tY = exp ζ(X,Y )

where ζ(X,Y ) belongs to the Lie algebra generated by [X,Y ] with:

ζ(X,Y ) = sX + tY +st

2[X,Y ] +

st2

12[[X,Y ], Y ]− s2t

12[[X,Y ], X]

−s2t2

24[X, [Y, [X,Y ]]] + · · · ,

the series converging for s, t small enough in the analytic case.

4. We haveexp tX exp εY exp−tX = exp η(X,Y )

with η(X,Y ) = ε∑k≥0

tk

k!adkX(Y ) and the series converging for ε, t small enough in the analytic case.

5. The ad-formula is:

exp tX ∗ Y =∑k≥0

tk

k!adkX(Y )

where the series is converging for t small enough.

Definition 4. A polysystem D is a family {Vi; i ∈ I} of vector fields. We denote by the same letter theassociated distribution, that is the mapping x 7→ Span{V (x);V ∈ D}. The distribution D is said to beinvolutive if [Vi, Vj ] ⊂ D, ∀Vi, Vj ∈ D.

Definition 5. Let D be a polysystem. We design by DAL the Lie algebra generated by D. By constructionthe associated distribution DAL is involutive. The Lie algebra DAL is constructed recursively as follows:

D1 = Span{D},D2 = Span{D1 + [D1, D1]},

. . . ,

Dk = Span{Dk−1 + [D1, Dk−1]}

and DAL = ∪k≥1Dk. If x ∈ M , we associate the following sequence of integers: nk(x) = dimDk(x).

2.1.1 Controllability with Piecewise Constant Controls

Definition 6. Consider a smooth system on M , given in local coordinates by

x(t) = f(x(t), u(t)), x(t) ∈ M, u(t) ∈ U ⊆ Rm (2.1)

The set of admissable controls u(·) is the set U of piecewise constant mappings. If x(t, x0, u) is the solution of2.1 associated to u(·) starting at at x(0) = x0, we denote by A(x0, T ) the accessibility set ∪u(·)∈Ux(T, x0, u)in time T and A(x0) the accessibility set ∪u(·)∈Ux(T, x0). The system is controllable if for each x0 ∈ M wehave A(x0) = M .

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Example 1. Consider the problem of a car parallel parking. The state of the car is given by its position and

rotation: q =

xyθ

∈ R2 × S1. The car can drive forward and backward at a bounded speed and can also

turn. This allows two controls u, v with (u, v) ∈ [−1, 1]× [−1, 1] to determine the possible trajectories of thevehicle. The system can thus be described as:x

y

θ

=

cos θsin θ0

u+

001

v (2.2)

Denote F1 =

cos θsin θ0

, F2 =

001

and F3 = [F1, F2]. The computation of F3 is as follows.

F3 = [F1, F2] =∂F1

∂qF2 −

∂F2

∂qF1

=

0 0 cos θ0 0 sin θ0 0 0

001

−

0 0 00 0 00 0 0

cos θsin θ0

=

− sin θcos θ0

As the rank of DAL(F1, F2, F3)(q) = 3, which is the same as the dimension of the manifold in which q resides(R2 × S1), we will see that this system is controllable. This is due to the Lie bracket allowing movement ina new direction (F3).

Definition 7. Consider a control system 2.1 on M . We can associate to this system the polysystem D ={f(·, u);u constant , u ∈ U}. We denote by ST (D) the set

ST (D) = {exp t1V1 · · · exp tkVk; k ∈ N, ti ≥ 0 andk∑

i=1

ti = T, Vi ∈ D}

and by S(D) the local semi-group: ∪T≥0ST (D). We denote by G(D) the local group generated by S(D), thatis

G(D) = {exp t1V1 · · · exp tkVk; k ∈ N, ti ∈ R, Vi ∈ D}.Properties.

1. The accessibility set from x0 in time T is:

A(x0, T ) = ST (D)(x0).

2. The accessibility set from x0 is the orbit of the local semi-group:

A(x0) = S(D)(x0).

Definition 8. We call the orbit of x0 the set O(x0) = G(D)(x0). The system is said to be weakly controllableif for every x0 ∈ M, O(x0) = M .

2.1.2 Integrating Distributions

Let D be a polysystem and DAL the Lie algebra generated by D. We consider the distribution ∆ : x 7→ DAL.It is an involutive distribution and the problem of integrating ∆ at a point x0 is to find a submanifold N ,containing x0 such thtat for each y ∈ N, TyN = ∆(y). It is a generalization of the Cauchy problem forintegrating a single vector field. Here, we are presenting two results:

1. If near x0 the rank of ∆ is constant, then we have the Frobenius Theorem which is a generalizationof the theorem of linearization of a smooth vector field X near a regular point.

2. If the rank is not constant but if the vector fields of the polysystem are real analytic, then the resultis still true. It was proved by Nagano-Sussmann.

The proofs of both are radically different.

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2.1.3 Frobenius Theorem

Theorem 1. (Frobenius Theorem) Assume that the rank of the distribution ∆ is constant near the pointx0 : rank∆ = p. Then there exists a local coordinate system x = (x1, · · · , xn) such that ∆ is generated by{ ∂∂x1 , · · · , ∂

∂xn }. In particular near x0 the integral manifolds are given by xi constant, for i = p+ 1, · · · , n.

Proof. The proof is standard and is a reccurence on p.If p = 1, then locally ∆ = RX where X ∈ V (M), X(x0) = 0. We use the linearization theorem for

ordinary differential equations.If p ≥ 2, then locally ∆ = Span{Y1, · · · , Yp}. We choose a coordinate system y = (y1, · · · , yn) centered

at x0 such that Y1 = ∂∂y1 . Consider the following p vector fields Z1, · · · , Zp of ∆ defined by

Z1 = Y1, Zk = Yk − (LYky1)Y1, 2 ≤ k ≤ p.

By construction LZky1 = 0 for 2 ≤ k ≤ p. Hence locally Zk is tangent for 2 ≤ k ≤ p to the submanifold

S : y1 = 0. Therefore there exist (p − 1) vector fields Vk on S which are the restriction of the vector fieldsZk to S. They define on S an involutive distribution ∆ of rank (p− 1). We use the reccurence assumptionwhich asserts that there exists on S a local coordinate system z = (z1, · · · , zn) such that

∆ = Span{V2, · · · , Vk} = Span{ ∂

∂z2, · · · , ∂

∂zp}.

We can define a local coordinate system centered at x0 by

x1 = y1, xi = zi, 2 ≤ i ≤ n.

We claim that ∆ = Span{Z1, · · · , Zp} coincides locally with the flat distribution Span{ ∂∂x1 , · · · , ∂

∂xn }. Toprove it, it is sufficient to check that

LZkxp+r = 0 for r ≥ 1 and 1 ≤ k ≤ p.

1. For k = 1 : since Z1 = ∂∂x1 , we have LZ1x

i = 0 for i ≥ 2.

2. For k ≥ 2 : first of all we observe that

∂

∂x1(LZk

xp+r) = LZ1(LZkxp+r).

Since LZ1xp+r = 0, we can write

∂

∂x1(LZk

xp+r) = L[Zk,Z1](xp+r)

and we know by construction that

[Z1, Zk] ∈ Span{Zj ; j ≥ 2}.

Hence we can write∂

∂x1(LZk

xp+r) =

p∑j=2

λj(LZjxp+r)

where the λj are scalar. It is a linear differential equation with respect to x1. For x1 = 0, we haveZk = Vk and by construction LVk

xp+r = 0. Since the solution of a linear system with values 0 atx1 = 0 is the identically zero solution, we have LZk

xp+r = 0.

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2.1.4 Nagano-Sussman Theorem

See [30]. When the rank condition is satisfied (rank∆ = constant) we get from the Frobenius theorem adescription of all the integral manifolds near x0. If we only need to construct the leaf passing through x0

the rank condition is clearly too strong. Indeed, if D = {X} is generated by a single vector field X, thereexists an integral curve through x0 which is locally Lipschitz. For a family of vector fields this result is stilltru if the vector fields are analytic.

Theorem 2. (Nagano-Sussman Theorem) Let D be a family of analytic vector fields near x0 ∈ M andlet p be the rank of ∆ : x 7→ DAL(x) at x0. Then through x0 there exists locally an integral manifold ofdimension p.

Proof. Let p be the rank of ∆ at x0. Then there exists p vector fields of DAL : X1, · · · , Xp such thatSpan{X1(x0), · · · , Xp(x0)} = ∆(x0). Consider the mapping

α : (t1, · · · , tp) 7→ exp t1X1 · · · exp tpXp(x0).

It is an immersion for (t1, · · · , tp) = (0, · · · , 0). Hence the image denoted by N is locally a submanifold ofdimension p. To prove that N is an integral manifold we must check that for each y ∈ N near x0, we haveTyN = ∆(y). It is a direct consequence of the equalities

DAL(exp tXi(x)) = d exp tXi(DAL(x)), i = 1, · · · , p

and x near x0, t small enough. To show that the previous equalities hold, let V (x) ∈ DAL(x) such thatV (x) = Y (x). By analycity and the ad-formula for t small enough we have

(d exp tXi)(Y (x)) =∑k≥0

tk

k!adkXi(Y )(exp tXi(x)).

Hence for t small enough, we have

(d exp tXi)(DAL(x)) ⊂ DAL(exp tXi(x)).

Changing t to −t we show the second inclusion.

2.1.5 C∞-Counter Example

To prove the previous theorem we use the following geometric property. Let X,Y be two analytic vectorfields and assume X(x0) = 0. From the ad-formula, if all the vector fields are adkX(Y ), k ≥ 0 are collinearto X at x0, then for t small enough the vector field Y is tangent to the integral curve exp tX(x0).

Hence is is easy to construct a C∞-counter example using flat C∞-mappings. Indeed take f : R 7→ R asmooth mapping such that f(x) = 0 for x ≤ 0 and f(x) = 0 for x > 0. Consider the two vector fields onR2 : X = ∂

∂x and Y = f(x) ∂∂y . At 0, DAL is of rank 1. Indeed, we have [X,Y ](x) = −f ′(x) ∂

∂y = 0 at 0 and

hence [X,Y ](0) = 0. The same is true for all high order Lie brackets. In this example the rank DAL is notconstant along exp tX(0), indeed for x > 0, the vector field Y is transverse to this vector field.

2.1.6 Nonlinear Controllability and Chow Theorem

Theorem 3. (Chow). Let D be a C∞-polysystem on M . We assume that for each x ∈ M,DAL(x = TxM).Then we have

G(D)(x) = G(DAL(x)) = M,

for each x ∈ M .

Proof. Since M is connected it is sufficient to prove the result locally. The proof is based on the BHC-formula. We assume M = R3 and D = {X,Y } with rank {X,Y, [X,Y ]} = 3 at x0; the generalization isstraightforward. Let λ be a real number and consider the mapping

φλ : (t1, t2, t3) 7→ expλX exp t3Y exp−λX exp t2Y exp t1X(x0).

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We prove that for λ small but nonzero, φλ is an immersion. Indeed using the BHC formula we have

φλ(t1, t2, t3) = exp(t1X + (t2 + t3)Y +λt32

[X,Y ] + · · · )(x0),

hence∂φλ

∂t1(0, 0, 0) = X(x0),

∂φλ

∂t2(0, 0, 0) = Y (x0),

∂φλ

∂t3(0, 0, 0) = Y (x0) +

λ

2[X,Y ](x0) + o(λ)

Since X,Y, [X,Y ] are linearly independent at x0, the rank of φλ at 0 is 3 for λ = 0 small enough.

Definition 9. The polysystem D is called weakly controllable if the orbit O(x) of G(D) is M for everyx ∈ M . The polysystem D is called controllable if the orbit A(x) of S(D) is M for every x ∈ M . Thepolysystem D is said symmetric if for every X ∈ D, we have −X ∈ D.

Example 2. Let D = { ∂∂x ,

∂∂y} on R2. Hence O(0) = R2 and A(0) = {x ≥ 0, y ≥ 0}. In general we have

that A(x) is a strict subset of O(x).

Corollary 1. Let D be a symmetric polysystem. Assume that

rank DAL = n (dimM) for every x.

Then D is controllable. In the analytic case this rank condition is also necessary.

Proof. Since D is symmetric, we have: orbit S(D) = orbit G(D). Then we apply the Chow theorem. In theanalytic case, we apply the Nagano-Sussmann theorem.

The symmetric case is the only case where we can conclude trivially that S(D) = G(D). In general theproblem to decide if a semi-group is transitive is difficult.

Nevertheless the following weaker result is true, see [31].

Proposition 2. Let D be a polysystem. If dim DAL = n (dimM) for every x ∈ M then for each neighborhoodV of x there exists a nonempty open set U contained in V ∩A(x).

Proof. Let x ∈ M . If dim M ≥ 1, there exists X1 ∈ D such that X1 = 0, otherwise we would have thatdim DAL(x) = 0. Consider the integral curve α1 : t 7→ exp tX1(x). If dim M ≥ 2, then there existsin every neighborhood V of x a point y ∈ M such that y = exp t1X1(x) and vector field X2 ∈ D suchthat X2 and X1 are not collinear at y, otherwise we would have dim DAL = 1. Consider the mappingα2 : (t1, t2) 7→ exp t2X2 exp t1X1(x). If dim M ≥ 3, then there exists in every neighborhood of y a vectorfield X3 transverse to the image α2. With this method we construct in every neighborhood V of x a mappingαn : (t1, · · · , tn) 7→ exp tnXn · · · exp t1X1(x) such that in a point z = αn(t

∗1, · · · , t∗n) of V , αn is an immersion.

This construction provides a nonempty open set U sontained in V ∩O(x).

2.1.7 Poisson Stability and Controllability

Definition 10. Let X be a C∞-vector field on M . The point x0 ∈ M is said Poisson stable if for every T > 0and every neighborhood V of x0 then there exists t1, t2 ≥ T such that exp t1X(x0) and exp−t2X(x0) ∈ V .The vector field X is called Poisson stable if the set of Poisson stable points is dense in M .

Theorem 4. (Poincare) [12] If M is a compact manifold with a volume form ω, each conservative vectorfield X is Poisson stable.

Proposition 3. Let D be a polysystem. Assume the following:

i) for every x ∈ M , rank DAL(x) = n (dimM);

ii) every vector field X ∈ D is Poisson stable.

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Then the system is controllable.

Proof. Here is an outline of the proof, see [22] for details. Let x, y ∈ M , we must show that there existsX1, · · · , Xk ∈ D and t1, · · · , tk > 0 such that

y = exp t1X1 · · · exp tkXk(x).

Since D satisfies the rank condition, we can apply Proposition 2 to D and −D to find the existence oftwo points x′, y′ and two open sets U and V such that x′ ∈ U, y′ ∈ V such that x′ can be steered usingXi ∈ D,−D to each point of U and each point of V can be steered to y′. To prove the proposition it issufficient to show that there exist two points x′′ ∈ U and y′′ ∈ V such that x′′ can be steered to y′′. Sincethe polysystem satisfies the rank condition, there exist p vector fields Y1, · · · , Yp and p nonzero (positive ornegative) real numbers si such that

y′ = exp s1Y1 · · · exp spYp(x′′).

In the previous sequence each element exp skYk corresponding to the negative time sk can be nearby replacedby an arc exp s′kYk using the Poisson-stability of Yk. The result follows.

2.1.8 Application

A first application of Proposition 3 is to construct many controllable polysystems on compact manifoldsusing Poincare theorem.

Example 3. Take M = G a compact Lie group and D a polysystem whose each vector field is a rightinvariant. Then the polysystem is controllable if and only if the rank condition is satisfied. In this case,observe that DAL is a Lie sub-algebra of g ≃ TeG and hence is finite dimensional. There exist algorithms tocompute DAL.

2.1.9 Controllability and Enlargement Technique

Let D be a polysystem satisfying the rank condition: rank DAL(x) = dimM for all x ∈ M . To study thecontrollability of such a polysystem, a powerful technique is the enlargement technique which was codifiedby Jurdjevic-Kupka [18]. The principle is simple, we enlarge D using operations which are not modifyingthe controllability of D. We shall briefly explain these operations. See also [17].

Lemma 1. Let D be a polysystem such that rank DAL(x) = dim M for all x. Then the polysystem D iscontrollable if and only if the adherence of S(D)(x) is M for every x ∈ M .

Proof. Use Proposition 2.

Definition 11. Let D,D′ be two polysystems satisfying the rank condition. We say that D and D′ areequivalent if for every x ∈ M : S(D)(x) = S(D′)(x). The union of all polysystem D′ equivalent to D iscalled the saturated of D and is denoted by sat D.

Clearly a polysystem D is controllable if and only if sat D is controllable. Now, we define the codifiedoperations.

Proposition 4. Let D be a polysystem, then the convex cone generated by sat D is equivalent to D.

Proof. Clearly if X ∈ D then λX ∈ sat D for every λ > 0 (reparameterization). Let X,Y ∈ D, using BHCformula we have ∏

n times

expt

nX exp

t

nY = exp(t(X + Y ) + o(

1

n)).

Taking the limit when n → +∞, we have X + Y ∈ sat D.

Proposition 5. Let X ∈ D and assume X Poisson stable, then −X ∈ D.

Proof. It is a consequence of the proof of Proposition 3.

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Proposition 6. If ±X,±Y ∈ D, then ±[X,Y ] ∈ sat D.

Proof. Apply the BHC formula.

Example 4. Consider a control system of form

x(t) = F0(x(t)) +m∑i=0

Fi(x(t))ui(t)

where the ui(t) are piecewise constant functions in {±1}. Let m = 1, F0 =

(x22

0

), F1 =

(01

), resulting in

x(t) =

(x22

0

)+

(01

)u(t).

Then, for the polysystem D = (F0, F1), the first terms of the Lie algebra DAL are:

[F0, F1] =∂F0

∂xF1 −

∂F1

∂xF0 =

(0 2x2

0 0

)(01

)−(0 00 0

)(x22

0

)=

(2x2

0

),

[[F0, F1], F1] =

(20

).

Then it is clear that dimDAL = 2, and the rank condition is satisfied. However, consider any trajectorybeginning with x2(0) > 0. Then the system is not controllable, as these trajectories can not move in thenegative x1 direction. This shows that the rank condition is necessary but not sufficient for controllability.

Definition 12. Let D be a polysystem on M . The normalizer N(D) of D is the set of diffeomorphisms φon M such that for every x ∈ M , φ(x) and φ−1(x) belong to the adherence of S(D)(x).

Proposition 7. Let D be a polysystem, X be an element of D and φ ∈ N(D). Then φ∗X belongs to satD.

Proof. Let Y = φ ∗X, then we have

y = (exp tY )(x) = φ ◦ exp tX ◦ φ−1(x)

Hence y belongs to the adherence of S(D)(x). The proposition follows.

Proposition 8. If D is a polysystem, then the closure of D for the topology of uniform convergence on thecompact sets belongs to sat D.

Proof. If Xn → X when n → +∞ for the above topology, then exp tXn 7→ exp tX when n → +∞ on eachcompact set.

Consider the linear system of Rn : x(t) = Ax(t) + Bu(t), u(t) ∈ Rp, A,B constant matrices. Then thelinear system is controllable if and only if the rank of R = [B, · · · , An−1B] is n. Indeed, the system canbe written x = Ax(t) +

∑pi=1 ui(t)bi and we introduce the polysystem D = {Ax +

∑pi=1 uibi;ui ∈ Rp}.

Computing we haveDAL = Ax⊕ ImR

and the rank is minimal at 0 and equals to the rank of R. Since the system is analytic, we must haverank R = n.

Let us prove the converse. Let i ∈ {1, · · · , p} then ±bi ∈ sat D. Indeed for every n ∈ N

1

n(Ax+Bu) ∈ sat D

and setting ui = nε, ε = ±1 we have

limn→+∞

1

n(A+ nεbi) = εbi ∈ sat D.

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Hence for every λ ∈ R, we haveexpλbi ∗Ax ∈ sat D.

Computing we obtain

expλbi ∗Ax =∑k≥0

λk

k!adkbi(Ax)

and adkbi(Ax) = 0 for k ≥ 2. Hence for λ = 0 we have

1

|λ|expλbi ∗Ax =

1

|λ|(Ax− λAbi) ∈ sat D.

Taking the limit when λ → ∞ we obtain ±Abi ∈ satD. Then we repeat the same operation, replacing bi byAbi. At the end we have

Span{bi, · · · , Akbi, · · · ; i = 1, · · · , p} = R ∈ sat D

The result is proved.

Proposition 9. Consider the following affine system on M :

x(t) = F0(x(t)) +

p∑i=1

ui(t)Fi(x(t)), ui(t) ∈ R.

Let D be the distributionx 7→ Span{F1(x), · · · , Fp(x)}.

If rank DAL = dimM , for all x ∈ M , then the system is controllable.

Proof. As before for every n ∈ N,ui = nε, ε = ±1, uj = 0 if j = i

1

n(F0 + nεFi) ∈ sat D

and by making n → +∞ we have ±Fi ∈ satD for i = 1, · · · , p. Applying Proposition 6 we have DAL ∈ satD.If DAL is of rank n, the system is controllable.

2.1.10 Evaluation of the Accessibility Set

The Baker-Campbell-Hausdorff formula can be used to make evaluation the accessibility set by constructingan approximation cone. In spirit it is similar to the idea pf Pontryagin maximum principle where weconstruct the first order Pontryagin cone. This was extensively used by Hermes [15] to get higher ordernecessary optimality conditions along a singular arc.

Definition 13. A rational polynomial is an expression of the form∑l

i=1 citqi , where l ∈ N, t > 0 small

enough, qi ∈ Q and ci ∈ R. It is called positive if ci ≥ 0 for all i = 1, · · · , l. Let X,Y ∈ V (M), D bethepolysystem {X,Y }, DAL the Lie algebra generated by D. We denote by E the set of germs of vector fieldsW such that there exists k ∈ N and rational polynomials r, · · · , rk, s1, · · · , sk : [0, ε] → R such that

exp rk(t)X exp sk(t)Y · · · exp r1(t)X exp s1(t)Y = exp(tW +O(tα))

with α > 1.

From the BHC formula, the set E is contained in DAL. We shall prove the following result.

Theorem 5. The set E is DAL

Optimal Control Theory Summer 2015

In order to prove this result we need several lemmas and the following two formulae

exp t1X exp t2W exp−t1X = exp

(t2(

n∑k=0

tk1k!adkX(W ) + o(tn1 ))

)(2.3)

exp t1X exp t2W = exp(t1X + t2W +t1t22

[X,W ] +t1t

22

12[[X,W ],W ]

− t21t212

[[X,W ], X]− t21t22

24[X, [W, [X,w]]] + · · · )

(2.4)

Lemma 2. The set E is convex.

Proof. Let λ ∈ [0, 1] and W1,W2 ∈ E . Then there exists k1, k2 ∈ N and rational polynomials such that

exp r1k1(t)X · · · exp s11(t)Y = exp(tW1 +O(tα))

exp r2k2(t)X · · · exp s21(t)Y = exp(tW2 +O(tα))

Hence we have

exp(r1k1(λt)X) · · · exp(s11(λt)Y ) exp(r2k2

(1− λ)t)X · · · exp(s21(1− λ)t)Y

= exp(λtW1 + (1− λ)tW2 +O(tα))

where α > 1. Hence λW1 + (1− λ)W2 ∈ E .

Lemma 3. The vector fields −X and −Y belong to E.

Proof. For α, β ∈ R we have

exp(αtX) exp(βtY ) = exp(t(αX + βY ) +O(t2)).

If we set α = −1, β = 0, we have −X ∈ E , and if we set α = 0, β = −1, we have −Y ∈ E .

Proof. (Theorem 5). We must show that if ±W ∈ E , then ±[X,W ] and ±[Y,W ] ∈ E . Since ±W ∈ E ,there exist rational polynomials such that

exp rk(t)X · · · exp s1(t)Y = exp(tW +O(tα))

exp r′k(t)X · · · exp s′1(t)Y = exp(−tW +O(tα))

with α > 1. Let β ∈ Q with 0 < β < 1 and αβ > 1. Then

exp(t1−βX) exp(tβW +O(tαβ)) exp(−t1−βX) exp(−tβW +O(tαβ))

= exp(tβW + t[X,W ] + · · · ) exp(−tβW +O(tαβ))

= exp(t[X,W ] + · · · ).

This proves that [X,W ] ∈ E . Hence by reccurrence we show that

±[adknY, [adkn−1X, [· · · , [adk1X,Y ] · · · ]

belongs to E . Since these Lie brackets are generating DAL the theorem is proved.

Acknowledgments

M. Chyba is partially supported by the National Science Foundation (NSF) Division of Mathematical Sci-ences, award #1109937.

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